The wavefunctions for the free-fermion part of the spectrum of the SUq(N) quantum spin models

We conjecture that the free-fermion part of the eigenspectrum observed recently for the SUq(N) Perk–Schultz spin chain Hamiltonian in a finite lattice with q = exp(iπ(N − 1)/N) is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous SUq(N − 1) vertex model which appears in the nested Bethe ansatz approach. We prove that this conjecture is valid for the case of the SUq(3) spin chain with periodic boundary condition. In this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model (q = exp(2iπ/3)), which permits us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free-fermion part of the eigenspectrum of the SUq(3) spin chain. Similarly, as in the known case of the SUq(2) case at q = exp(i2π/3) our numerical and analytical studies induce some conjectures for special rates of correlation functions.